To solve the equation [tex]\(7^{4x} = 5\)[/tex] for [tex]\(x\)[/tex], we will follow a step-by-step process using logarithms. Here is the detailed solution:
1. Rewrite the equation using logarithms:
We start by taking the natural logarithm (or any logarithm) on both sides of the equation to make use of the logarithm properties.
[tex]\[
\ln(7^{4x}) = \ln(5)
\][/tex]
2. Apply the power rule of logarithms:
The power rule states that [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex]. Applying this rule,
[tex]\[
4x \cdot \ln(7) = \ln(5)
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Now, isolate [tex]\(x\)[/tex] by dividing both sides by [tex]\(4 \cdot \ln(7)\)[/tex]:
[tex]\[
x = \frac{\ln(5)}{4 \cdot \ln(7)}
\][/tex]
4. Calculate the logarithms and solve:
We need to compute [tex]\(\ln(5)\)[/tex] and [tex]\(\ln(7)\)[/tex]. Using a calculator:
[tex]\[
\ln(5) \approx 1.60944
\][/tex]
[tex]\[
\ln(7) \approx 1.94591
\][/tex]
Substitute these values back into the equation:
[tex]\[
x = \frac{1.60944}{4 \times 1.94591}
\][/tex]
5. Perform the division:
First, find the value of [tex]\(4 \times \ln(7)\)[/tex]:
[tex]\[
4 \times 1.94591 \approx 7.78364
\][/tex]
Then divide:
[tex]\[
x = \frac{1.60944}{7.78364} \approx 0.2068
\][/tex]
6. Round the result to the nearest hundredth:
The final value, rounded to the nearest hundredth, is:
[tex]\[
x \approx 0.21
\][/tex]
Therefore, the solution to the equation [tex]\(7^{4x} = 5\)[/tex] is [tex]\( x \approx 0.21 \)[/tex].
